If your router has a USB port, it probably has an option to share a HDD over the network. Just put the HDD into a USB enclosure and plug it in. Then configure the router for shared storage.
It won't be very fast - older routers typically only achieved 10-15 MB/s with a HDD hooked up this way. Newer ones usually manage 30-40 MB/s. But that should be fast for basic file sharing and backups. The first backup will take forever, but if you make sure subsequent backups are incremental or differential it won't take anywhere near as much time.
After that, what you want is called a NAS - network attached storage. Those typically are able to hit 30-60 MB/s transfer speeds for the cheap ones, 80-120 MB/s for the expensive ones. I discourage using an old computer for this purpose because a NAS is almost always left on 24/7. Older computers typically burned close to 100 Watts at idle, while a dedicated NAS will burn closer to 15-20 Watts. If you pay the average U.S. electricity price, over a year that extra 80 Watts will cost you $80. And in a few years you've paid enough extra in electricity to have just bought the NAS to begin with. The only "old computer" I recommend for this purpose is an old laptop (those typically burn about 8-15 Watts).